Study on Delaunay Tessellations of 1-Irregular Cuboids for 3D Mixed Element Meshes
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Unit 6 Visualising Solid Shapes(Final)
• 3D shapes/objects are those which do not lie completely in a plane. • 3D objects have different views from different positions. • A solid is a polyhedron if it is made up of only polygonal faces, the faces meet at edges which are line segments and the edges meet at a point called vertex. • Euler’s formula for any polyhedron is, F + V – E = 2 Where F stands for number of faces, V for number of vertices and E for number of edges. • Types of polyhedrons: (a) Convex polyhedron A convex polyhedron is one in which all faces make it convex. e.g. (1) (2) (3) (4) 12/04/18 (1) and (2) are convex polyhedrons whereas (3) and (4) are non convex polyhedron. (b) Regular polyhedra or platonic solids: A polyhedron is regular if its faces are congruent regular polygons and the same number of faces meet at each vertex. For example, a cube is a platonic solid because all six of its faces are congruent squares. There are five such solids– tetrahedron, cube, octahedron, dodecahedron and icosahedron. e.g. • A prism is a polyhedron whose bottom and top faces (known as bases) are congruent polygons and faces known as lateral faces are parallelograms (when the side faces are rectangles, the shape is known as right prism). • A pyramid is a polyhedron whose base is a polygon and lateral faces are triangles. • A map depicts the location of a particular object/place in relation to other objects/places. The front, top and side of a figure are shown. Use centimetre cubes to build the figure. -
Uniform Panoploid Tetracombs
Uniform Panoploid Tetracombs George Olshevsky TETRACOMB is a four-dimensional tessellation. In any tessellation, the honeycells, which are the n-dimensional polytopes that tessellate the space, Amust by definition adjoin precisely along their facets, that is, their ( n!1)- dimensional elements, so that each facet belongs to exactly two honeycells. In the case of tetracombs, the honeycells are four-dimensional polytopes, or polychora, and their facets are polyhedra. For a tessellation to be uniform, the honeycells must all be uniform polytopes, and the vertices must be transitive on the symmetry group of the tessellation. Loosely speaking, therefore, the vertices must be “surrounded all alike” by the honeycells that meet there. If a tessellation is such that every point of its space not on a boundary between honeycells lies in the interior of exactly one honeycell, then it is panoploid. If one or more points of the space not on a boundary between honeycells lie inside more than one honeycell, the tessellation is polyploid. Tessellations may also be constructed that have “holes,” that is, regions that lie inside none of the honeycells; such tessellations are called holeycombs. It is possible for a polyploid tessellation to also be a holeycomb, but not for a panoploid tessellation, which must fill the entire space exactly once. Polyploid tessellations are also called starcombs or star-tessellations. Holeycombs usually arise when (n!1)-dimensional tessellations are themselves permitted to be honeycells; these take up the otherwise free facets that bound the “holes,” so that all the facets continue to belong to two honeycells. In this essay, as per its title, we are concerned with just the uniform panoploid tetracombs. -
Hexagonal Antiprism Tetragonal Bipyramid Dodecahedron
Call List hexagonal antiprism tetragonal bipyramid dodecahedron hemisphere icosahedron cube triangular bipyramid sphere octahedron cone triangular prism pentagonal bipyramid torus cylinder squarebased pyramid octagonal prism cuboid hexagonal prism pentagonal prism tetrahedron cube octahedron square antiprism ellipsoid pentagonal antiprism spheroid Created using www.BingoCardPrinter.com B I N G O hexagonal triangular squarebased tetrahedron antiprism cube prism pyramid tetragonal triangular pentagonal octagonal cube bipyramid bipyramid bipyramid prism octahedron Free square dodecahedron sphere Space cuboid antiprism hexagonal hemisphere octahedron torus prism ellipsoid pentagonal pentagonal icosahedron cone cylinder prism antiprism Created using www.BingoCardPrinter.com B I N G O triangular pentagonal triangular hemisphere cube prism antiprism bipyramid pentagonal hexagonal tetragonal torus bipyramid prism bipyramid cone square Free hexagonal octagonal tetrahedron antiprism Space antiprism prism squarebased dodecahedron ellipsoid cylinder octahedron pyramid pentagonal icosahedron sphere prism cuboid spheroid Created using www.BingoCardPrinter.com B I N G O cube hexagonal triangular icosahedron octahedron prism torus prism octagonal square dodecahedron hemisphere spheroid prism antiprism Free pentagonal octahedron squarebased pyramid Space cube antiprism hexagonal pentagonal triangular cone antiprism cuboid bipyramid bipyramid tetragonal cylinder tetrahedron ellipsoid bipyramid sphere Created using www.BingoCardPrinter.com B I N G O -
Convex Polytopes and Tilings with Few Flag Orbits
Convex Polytopes and Tilings with Few Flag Orbits by Nicholas Matteo B.A. in Mathematics, Miami University M.A. in Mathematics, Miami University A dissertation submitted to The Faculty of the College of Science of Northeastern University in partial fulfillment of the requirements for the degree of Doctor of Philosophy April 14, 2015 Dissertation directed by Egon Schulte Professor of Mathematics Abstract of Dissertation The amount of symmetry possessed by a convex polytope, or a tiling by convex polytopes, is reflected by the number of orbits of its flags under the action of the Euclidean isometries preserving the polytope. The convex polytopes with only one flag orbit have been classified since the work of Schläfli in the 19th century. In this dissertation, convex polytopes with up to three flag orbits are classified. Two-orbit convex polytopes exist only in two or three dimensions, and the only ones whose combinatorial automorphism group is also two-orbit are the cuboctahedron, the icosidodecahedron, the rhombic dodecahedron, and the rhombic triacontahedron. Two-orbit face-to-face tilings by convex polytopes exist on E1, E2, and E3; the only ones which are also combinatorially two-orbit are the trihexagonal plane tiling, the rhombille plane tiling, the tetrahedral-octahedral honeycomb, and the rhombic dodecahedral honeycomb. Moreover, any combinatorially two-orbit convex polytope or tiling is isomorphic to one on the above list. Three-orbit convex polytopes exist in two through eight dimensions. There are infinitely many in three dimensions, including prisms over regular polygons, truncated Platonic solids, and their dual bipyramids and Kleetopes. There are infinitely many in four dimensions, comprising the rectified regular 4-polytopes, the p; p-duoprisms, the bitruncated 4-simplex, the bitruncated 24-cell, and their duals. -
Step 1: 3D Shapes
Year 1 – Autumn Block 3 – Shape Step 1: 3D Shapes © Classroom Secrets Limited 2018 Introduction Match the shape to its correct name. A. B. C. D. E. cylinder square- cuboid cube sphere based pyramid © Classroom Secrets Limited 2018 Introduction Match the shape to its correct name. A. B. C. D. E. cube sphere square- cylinder cuboid based pyramid © Classroom Secrets Limited 2018 Varied Fluency 1 True or false? The shape below is a cone. © Classroom Secrets Limited 2018 Varied Fluency 1 True or false? The shape below is a cone. False, the shape is a triangular-based pyramid. © Classroom Secrets Limited 2018 Varied Fluency 2 Circle the correct name of the shape below. cylinder pyramid cuboid © Classroom Secrets Limited 2018 Varied Fluency 2 Circle the correct name of the shape below. cylinder pyramid cuboid © Classroom Secrets Limited 2018 Varied Fluency 3 Which shape is the odd one out? © Classroom Secrets Limited 2018 Varied Fluency 3 Which shape is the odd one out? The cuboid is the odd one out because the other shapes are cylinders. © Classroom Secrets Limited 2018 Varied Fluency 4 Follow the path of the cuboids to make it through the maze. Start © Classroom Secrets Limited 2018 Varied Fluency 4 Follow the path of the cuboids to make it through the maze. Start © Classroom Secrets Limited 2018 Reasoning 1 The shapes below are labelled. Spot the mistake. A. B. C. cone cuboid cube Explain your answer. © Classroom Secrets Limited 2018 Reasoning 1 The shapes below are labelled. Spot the mistake. A. B. C. cone cuboid cube Explain your answer. -
Crystal Chemical Relations in the Shchurovskyite Family: Synthesis and Crystal Structures of K2cu[Cu3o]2(PO4)4 and K2.35Cu0.825[Cu3o]2(PO4)4
crystals Article Crystal Chemical Relations in the Shchurovskyite Family: Synthesis and Crystal Structures of K2Cu[Cu3O]2(PO4)4 and K2.35Cu0.825[Cu3O]2(PO4)4 Ilya V. Kornyakov 1,2 and Sergey V. Krivovichev 1,3,* 1 Department of Crystallography, Institute of Earth Sciences, St. Petersburg State University, University Emb. 7/9, 199034 Saint-Petersburg, Russia; [email protected] 2 Laboratory of Nature-Inspired Technologies and Environmental Safety of the Arctic, Kola Science Centre, Russian Academy of Science, Fesmana 14, 184209 Apatity, Russia 3 Nanomaterials Research Center, Federal Research Center–Kola Science Center, Russian Academy of Sciences, Fersmana Str. 14, 184209 Apatity, Russia * Correspondence: [email protected] Abstract: Single crystals of two novel shchurovskyite-related compounds, K2Cu[Cu3O]2(PO4)4 (1) and K2.35Cu0.825[Cu3O]2(PO4)4 (2), were synthesized by crystallization from gaseous phase and structurally characterized using single-crystal X-ray diffraction analysis. The crystal structures of both compounds are based upon similar Cu-based layers, formed by rods of the [O2Cu6] dimers of oxocentered (OCu4) tetrahedra. The topologies of the layers show both similarities and differences from the shchurovskyite-type layers. The layers are connected in different fashions via additional Cu atoms located in the interlayer, in contrast to shchurovskyite, where the layers are linked by Ca2+ cations. The structures of the shchurovskyite family are characterized using information-based Citation: Kornyakov, I.V.; structural complexity measures, which demonstrate that the crystal structure of 1 is the simplest one, Krivovichev, S.V. Crystal Chemical whereas that of 2 is the most complex in the family. -
A Method for Predicting Multivariate Random Loads and a Discrete Approximation of the Multidimensional Design Load Envelope
International Forum on Aeroelasticity and Structural Dynamics IFASD 2017 25-28 June 2017 Como, Italy A METHOD FOR PREDICTING MULTIVARIATE RANDOM LOADS AND A DISCRETE APPROXIMATION OF THE MULTIDIMENSIONAL DESIGN LOAD ENVELOPE Carlo Aquilini1 and Daniele Parisse1 1Airbus Defence and Space GmbH Rechliner Str., 85077 Manching, Germany [email protected] Keywords: aeroelasticity, buffeting, combined random loads, multidimensional design load envelope, IFASD-2017 Abstract: Random loads are of greatest concern during the design phase of new aircraft. They can either result from a stationary Gaussian process, such as continuous turbulence, or from a random process, such as buffet. Buffet phenomena are especially relevant for fighters flying in the transonic regime at high angles of attack or when the turbulent, separated air flow induces fluctuating pressures on structures like wing surfaces, horizontal tail planes, vertical tail planes, airbrakes, flaps or landing gear doors. In this paper a method is presented for predicting n-dimensional combined loads in the presence of massively separated flows. This method can be used both early in the design process, when only numerical data are available, and later, when test data measurements have been performed. Moreover, the two-dimensional load envelope for combined load cases and its discretization, which are well documented in the literature, are generalized to n dimensions, n ≥ 3. 1 INTRODUCTION The main objectives of this paper are: 1. To derive a method for predicting multivariate loads (but also displacements, velocities and accelerations in terms of auto- and cross-power spectra, [1, pp. 319-320]) for aircraft, or parts of them, subjected to a random excitation. -
41 Three Dimensional Shapes
41 Three Dimensional Shapes Space figures are the first shapes children perceive in their environment. In elementary school, children learn about the most basic classes of space fig- ures such as prisms, pyramids, cylinders, cones and spheres. This section concerns the definitions of these space figures and their properties. Planes and Lines in Space As a basis for studying space figures, first consider relationships among lines and planes in space. These relationships are helpful in analyzing and defining space figures. Two planes in space are either parallel as in Figure 41.1(a) or intersect as in Figure 41.1(b). Figure 41.1 The angle formed by two intersecting planes is called the dihedral angle. It is measured by measuring an angle whose sides line in the planes and are perpendicular to the line of intersection of the planes as shown in Figure 41.2. Figure 41.2 Some examples of dihedral angles and their measures are shown in Figure 41.3. 1 Figure 41.3 Two nonintersecting lines in space are parallel if they belong to a common plane. Two nonintersecting lines that do not belong to the same plane are called skew lines. If a line does not intersect a plane then it is said to be parallel to the plane. A line is said to be perpendicular to a plane at a point A if every line in the plane through A intersects the line at a right angle. Figures illustrating these terms are shown in Figure 41.4. Figure 41.4 Polyhedra To define a polyhedron, we need the terms ”simple closed surface” and ”polygonal region.” By a simple closed surface we mean any surface with- out holes and that encloses a hollow region-its interior. -
Volume 75 (2019)
Acta Cryst. (2019). B75, doi:10.1107/S2052520619010047 Supporting information Volume 75 (2019) Supporting information for article: Lanthanide coordination polymers based on designed bifunctional 2-(2,2′:6′,2″-terpyridin-4′-yl)benzenesulfonate ligand: syntheses, structural diversity and highly tunable emission Yi-Chen Hu, Chao Bai, Huai-Ming Hu, Chuan-Ti Li, Tian-Hua Zhang and Weisheng Liu Acta Cryst. (2019). B75, doi:10.1107/S2052520619010047 Supporting information, sup-1 Table S1 Continuous Shape Measures (CShMs) of the coordination geometry for Eu3+ ions in 1- Eu. Label Symmetry Shape 1-Eu EP-9 D9h Enneagon 33.439 OPY-9 C8v Octagonal pyramid 22.561 HBPY-9 D7h Heptagonal bipyramid 15.666 JTC-9 C3v Johnson triangular cupola J3 15.263 JCCU-9 C4v Capped cube J8 10.053 CCU-9 C4v Spherical-relaxed capped cube 9.010 JCSAPR-9 C4v Capped square antiprism J10 2.787 CSAPR-9 C4v Spherical capped square antiprism 1.930 JTCTPR-9 D3h Tricapped trigonal prism J51 3.621 TCTPR-9 D3h Spherical tricapped trigonal prism 2.612 JTDIC-9 C3v Tridiminished icosahedron J63 12.541 HH-9 C2v Hula-hoop 9.076 MFF-9 Cs Muffin 1.659 Acta Cryst. (2019). B75, doi:10.1107/S2052520619010047 Supporting information, sup-2 Table S2 Continuous Shape Measures (CShMs) of the coordination geometry for Ln3+ ions in 2- Er, 4-Tb, and 6-Eu. Label Symmetry Shape 2-Er 4-Tb 6-Eu Er1 Er2 OP-8 D8h Octagon 31.606 31.785 32.793 31.386 HPY-8 C7v Heptagonal pyramid 23.708 24.442 23.407 23.932 HBPY-8 D6h Hexagonal bipyramid 17.013 13.083 12.757 15.881 CU-8 Oh Cube 11.278 11.664 8.749 11.848 -
Optimization of Tensegrity Lattice with Truncated Octahedral Units
OptimizationM. Ohsaki, J. Y. of Zhang, tensegrity K. Kogiso lattice andwith J. truncated J. Rimoli octahedral units Proceedings of the IASS Annual Symposium 2019 – Structural Membranes 2019 Form and Force 7 – 10 October 2019, Barcelona, Spain C. Lázaro, K.-U. Bletzinger, E. Oñate (eds.) Optimization of tensegrity lattice with truncated octahedral units Makoto OHSAKI∗a, Jingyao ZHANGa, Kosuke KOGISOa, Julian J. RIMOLIc *aDepartment of Architecture and Architectural Engineering, Kyoto University Kyoto-Daigaku Katsura, Nishikyo, Kyoto 615-8540, Japan [email protected] b School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA Abstract An optimization method is presented for tensegrity lattices composed of eight truncated octahedral units. The stored strain energy under specified forced vertical displacement is maximized under constraint on the structural material volume. The nonlinear behavior of bars allowing buckling is modeled as a bi- linear elastic material. As a result of optimization, a flexible structure with degrading vertical stiffness is obtained to be applicable to an energy absorption device or a vertical isolation system. Keywords: tensegrity, optimization, strain energy, truncated octahedral unit 1. Introduction Tensegrity structures are a class of self-standing pin jointed structures consisting of thin bars (struts) and cables stiffened by applying prestresses to their members [1, 3]. Because the bars are not continuously connected, i.e., they are topologically isolated from each other, these structures are usually too flexible to be used as load-resisting components of structures in mechanical and civil engineering applications. However, by exploiting their flexibility, tensegrity structures can be utilized as devices for reducing impact forces, isolating dynamic forces, and absorbing energy, to name a few applications [5]. -
How Many Times Can the Volume of a Convex Polyhedron Be Increased By
How many times can the volume of a convex polyhedron be increased by isometric deformations? Victor Alexandrov March 1, 2017 Abstract We prove that the answer to the question of the title is ‘as many times as you want.’ More precisely, given any constant c> 0, we construct two oblique triangular bipyramids, P and Q, such that P is convex, Q is nonconvex and intrinsically isometric to P , and vol Q > c·vol P > 0. Mathematics Subject Classification (2010): 52B10; 51M20; 52A15; 52B60; 52C25; 49Q10 Key words: Euclidean space, convex polyhedron, bipyramid, intrinsic metric, intrinsic isome- try, volume increasing deformation 1. Introduction. According to the classical theorem by A.L. Cauchy and A.D. Alexandrov, two compact boundary-free convex polyhedral surfaces in Euclidean 3-space are necessarily con- gruent as soon as they are isometric in their intrinsic metrics, see, e. g., [1]. Obviously, this is not true if at least one of the surfaces is nonconvex. In [4], the authors studied isometric immersions of polyhedral surfaces and, among other things, proved that there exists a compact boundary-free convex polyhedral surface allowing another isometric immersion which, being a nonconvex poly- hedral surface, encloses a larger volume than that enclosed by the original convex surface. This amazing existence theorem has gained new significance after the remarkable contribution of D.D. Bleecker, who explicitly built volume increasing isometric deformations of the surfaces of the Pla- tonic solids, see [3]. For example, he has shown that the surface of a regular tetrahedron can be isometrically deformed in such a way as to enlarge the enclosed volume by 37.7%. -
Unit Origami: Star-Building on Deltahedra Heidi Burgiel Bridgewater State University, [email protected]
Bridgewater State University Virtual Commons - Bridgewater State University Mathematics Faculty Publications Mathematics Department 2015 Unit Origami: Star-Building on Deltahedra Heidi Burgiel Bridgewater State University, [email protected] Virtual Commons Citation Burgiel, Heidi (2015). Unit Origami: Star-Building on Deltahedra. In Mathematics Faculty Publications. Paper 47. Available at: http://vc.bridgew.edu/math_fac/47 This item is available as part of Virtual Commons, the open-access institutional repository of Bridgewater State University, Bridgewater, Massachusetts. Proceedings of Bridges 2015: Mathematics, Music, Art, Architecture, Culture Unit Origami: Star-Building on Deltahedra Heidi Burgiel Dept. of Mathematics, Bridgewater State University 24 Park Avenue, Bridgewater, MA 02325, USA [email protected] Abstract This workshop provides instructions for folding the star-building unit – a modification of the Sonobe module for unit origami. Geometric questions naturally arise during this process, ranging in difficulty from middle school to graduate levels. Participants will learn to fold and assemble star-building units, then explore the structure of the eight strictly convex deltahedra. Introduction Many authors and educators have used origami, the art of paper folding, to provide concrete examples moti- vating mathematical problem solving. [3, 2] In unit origami, multiple sheets are folded and combined to form a whole; the Sonobe unit is a classic module in this art form. This workshop describes the construction and assembly of the star-building unit1, highlights a small selection of the many geometric questions motivated by this process, and introduces participants to the strictly convex deltahedra. Proficiency in geometry and origami is not required to enjoy this event. About the Unit Three Sonobe units interlock to form a pyramid with an equilateral triangular base as shown in Figure 1.